Average Error: 19.5 → 0.6
Time: 20.3s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -248713679.72383326:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{elif}\;z \le 8.728612712058619 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525\right)}{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}
↓
\begin{array}{l}
\mathbf{if}\;z \le -248713679.72383326:\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\

\mathbf{elif}\;z \le 8.728612712058619 \cdot 10^{-20}:\\
\;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525\right)}{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r17981847 = x;
        double r17981848 = y;
        double r17981849 = z;
        double r17981850 = 0.0692910599291889;
        double r17981851 = r17981849 * r17981850;
        double r17981852 = 0.4917317610505968;
        double r17981853 = r17981851 + r17981852;
        double r17981854 = r17981853 * r17981849;
        double r17981855 = 0.279195317918525;
        double r17981856 = r17981854 + r17981855;
        double r17981857 = r17981848 * r17981856;
        double r17981858 = 6.012459259764103;
        double r17981859 = r17981849 + r17981858;
        double r17981860 = r17981859 * r17981849;
        double r17981861 = 3.350343815022304;
        double r17981862 = r17981860 + r17981861;
        double r17981863 = r17981857 / r17981862;
        double r17981864 = r17981847 + r17981863;
        return r17981864;
}
↓
double f(double x, double y, double z) {
        double r17981865 = z;
        double r17981866 = -248713679.72383326;
        bool r17981867 = r17981865 <= r17981866;
        double r17981868 = 0.0692910599291889;
        double r17981869 = y;
        double r17981870 = 0.07512208616047561;
        double r17981871 = r17981869 / r17981865;
        double r17981872 = x;
        double r17981873 = fma(r17981870, r17981871, r17981872);
        double r17981874 = fma(r17981868, r17981869, r17981873);
        double r17981875 = 8.728612712058619e-20;
        bool r17981876 = r17981865 <= r17981875;
        double r17981877 = r17981868 * r17981865;
        double r17981878 = 0.4917317610505968;
        double r17981879 = r17981877 + r17981878;
        double r17981880 = r17981865 * r17981879;
        double r17981881 = 0.279195317918525;
        double r17981882 = r17981880 + r17981881;
        double r17981883 = r17981869 * r17981882;
        double r17981884 = 3.350343815022304;
        double r17981885 = 6.012459259764103;
        double r17981886 = r17981865 + r17981885;
        double r17981887 = r17981865 * r17981886;
        double r17981888 = r17981884 + r17981887;
        double r17981889 = r17981883 / r17981888;
        double r17981890 = r17981872 + r17981889;
        double r17981891 = r17981876 ? r17981890 : r17981874;
        double r17981892 = r17981867 ? r17981874 : r17981891;
        return r17981892;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.5
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.652456675:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -248713679.72383326 or 8.728612712058619e-20 < z

    1. Initial program 38.1

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

    2. Simplified31.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}, \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), x\right)}\]

    3. Taylor expanded around inf 1.1

      \[\leadsto \color{blue}{x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291889 \cdot y\right)}\]

    4. Simplified1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)}\]

    if -248713679.72383326 < z < 8.728612712058619e-20

    1. Initial program 0.1

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -248713679.72383326:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{elif}\;z \le 8.728612712058619 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525\right)}{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))